Pacific Journal of Mathematics
WEIERSTRASS POINTS OF PLANE DOMAINS
NEWTON SEYMOUR HAWLEY
Vol. 22, No. 2 February 1967 PACIFIC JOURNAL OF MATHEMATICS Vol. 22, No. 2, 1967
WEIERSTRASS POINTS OF PLANE DOMAINS
N. S. HAWLEY
In this paper examples will be given of plane domains which have interior points as Weierstrass points.
The notion of a Weierstrass point of a Riemann surface (or an algebraic curve) is an old one, having been introduced more than a century ago. Surprisingly enough, some of the simplest and most imme- diate questions concerning Weierstrass points remain unanswered. A beautiful account of the fundamental theory of Weierstrass points can be found in [2]. Nevertheless, we shall, for the sake of convenience, re- capitulate some of the basic facts concerning Weierstrass points. Let & be a closed Riemann surface of genus p > 1 and let
ul9 ---yUp be a basis of linearly independent Abelian integrals of the first kind on &. Consider
lu, duΌ dz dza n
p p d u1 d up
defined in the neighborhood Ua of the local uniformizer za. If Ua [Ί Uβ
is not empty and zβ is the uniformizer of Uβ, then in the neighborhood
Ua n Uβ we have
Wa = Wβ.
Thus {Wa} defines an everywhere finite differential of dimension P(P + 1)
on &. (Some authors refer to the degree of a differential rather than its dimension, others speak of its order. See [4].) Such a differential will have
(2p - = (P - 1)P(P
zeros (counted with proper multiplicity). These zeros are the Weierst- rass points of &. They do not depend on the particular basis
251 252 N. S. HAWLEY
uu - — >up chosen but are a fixed set of points of ^,
(p - l)p(p + 1) in number. In order to introduce another basic (and defining) property of Weierstrass points, we consider the following problem : for a given point q e & let us try to construct on & a function which is complex analytic and regular everywhere on & except at q and at q has a pole of order n. For all but a finite number of points, viz, all but the Weierstrass points we must have n ^ p + 1, but at each Weierstrass point we may choose an n S P We shall return to this interpretation later. The connection between the two definitions is easily established, e.g. see [2]. There is still a third fundamental definition of Weierstrass points which we shall use. This definition employs the " Noether mapping " υ of a nonhyperelliptic Riemann surface & of genus p into P^C), the complex protective space of dimension p — 1 (see [1]), and can only be employed if & is nonhyperelliptic. The mapping υ is accom- plished by selecting a basis of Abelian differentials of the first kind
on & and considering them as homogeneous coordinates in Pί?_1(C). Then u{&), the image of & in P^C), is a nonsingular curve of degree 2p — 2. The Weierstrass points of & are those point of υ (&) at which the osculating hyperplane is hyperosculating, and the degree of the hyper-osculation is the order of the Weierstrass point (see [2]). So, for example, if & is of genus 3 and nonhyperelliptic then p — 1 = 2 and υ{&) is a plane curve of degree 4 and the Weierstrass points are the inflection points of υ(&). Actually, what we are going to consider are Weierstrass points on plane domains, and we have only defined them so far on compact Riemann surfaces. But these definitions can be extended to plane domains by employing a technique due to Schottky, viz. the technique of doubling a plane domain (see [5] and [4]). We consider here only domains of finite connectivity which are bounded by analytic Jordan curves. Thus, if 2$ is a plane domain with p + 1 boundary curves, its double is a compact Riemann surface <%} of genus p. £& and its boundary curves are contained in <% as "half" of the Riemann surface the other " half " i^* has the same